ON SUBOBJECTS AND IMAGES IN CATEGORIES Oswald Wyler and Hans Ehrbar Report 68-34 November/ 1968 University Libraries Carnegie Mellon University Pittsburgh PA 15213-3890 ON SUBOBJECTS AND IMAGES IN CATEGORIES by Hans Ehrbar and Oswald Wyler* 1. Introduction Subobjects and images in categories have been defined in many ways. MacLane [9]] proposed an axiomatic theory of "bicategories" which was simplified by Isbell [5J. Grothendieck [41 defined subobjects (sous-trues) as equivalence classes of monomorphisms and suggested a definition of images which many authors have adopted (see e.g. CIO; I.10J). This works well in algebra, but not in general topology. Isbell £&), Jurchescu and Lascu £7j , Sonner fllj and others have suggested categorical remedies for this situation. No "absolute" definition of subobjects and images in a category has been proposed which is adequate for all situations. Moreover, in some situations, e.g. in general topology and in the theory of partial algebras, several reason- able definitions of subobjects are possible. Thus a "relative" theory of sub- objects and images is needed. In the present paper, we define .J-subobjects and ^-images in a category £ for an arbitrary class J, of morphisms of £ . Our definitions are equivalent to those of Grothendieck [4~\ if J. is the class of • Research supported in part by N.S.F. Grant GP-8773. HUNT LIBRARY MHNE6IE-MELL0N UNIVERSITY all monomorphisms of £ . We introduce a new concept, strong .J-images, and we study J^images and strong J.-images, and the resulting image functors, in sec- tions 3-5. Isbell's theory [$\ is generalized in section 6. In sections 7 - 8f we define and study direct and inverse images as strictly dual concepts, and we discuss briefly the resulting pseudofunctors and fibrations. Examples and com- ments are given in section 9. Further developments and applications have been obtained by each of us and will be published separately. One feature of our theory is that we do not require images to be monomorphic. There are several reasons for this. For instance, coimages in operational cate- gories need not be epimorphic, and to require monomorphic images would almost destroy applications to reflective subcategories ([2; cap. 2} and [&\). th m We shall refer by m.n to the n item of section ra, and the symbol H will denote the end or absence of a proof. The notations introduced in section 2 will be used throughout the paper. 2. Preliminaries Throughout this paper, let £ be a category and £ a class of morphisms of £ . From 3.5 on, we shall require J. to be left transportable (2.6). 2.1. We write composition in £ from left to right, so that f g means: first f , then g . We identify £ with its class of morphisms, and we denote by J£J the class of all objects of £ . We often identify an object A 6-(£| with the identity morphisra id A £ £ . We denote domain and codomain of a mor- phism f £ £ by f DQ and f ^ 9 so that (f DQ) f » f (f 1^) * f in £ . We write functors and natural transformations as right operators, with composi- tions from left to right. Id £ denotes the identity functor on £ and id T the identity natural transformation, with A (id T) « id (A T) for an object A £ P denotes the dual category of £ . 2 2.2. We denote by £- the category with morphisms of £ as objects and 2 commutative squares in £ as morphisms. A morphism of C7 9 from f £. £ to g £C , is a quadruple (f*u,v,g) of morphisms of £ such that f v « u g 2 in £ . Composition in Of* is given by (f,u,v,g)(g,u\vf,h) « (f,u u\v vf,h) . We write (u,v) : f—>g , and sometimes just (u,v) , for (f,u,v,g) . 2 We define a domain functor D and a codomain functor D. from Cr to C _ o : 1 —. — by putting (f ,u,v,g) DQ * u , (f ,u,v,g) D1 » v for a morphism (f,u,v,g) of O7 . This agrees with the notation of 2.1 for an 2 object f of £- 9 i.e. a morphism f of £ . 2 2 We denote by £rGl3 the full subcategory of £- with morphisms in J[ as 2 2 its objects, and by I : £"£J3 —> £T the inclusion functor. A morphism of ^s a commutative square in £ with two opposite sides from J^ . The dual category of £r[ij is obtained by a variant of the usual reversal of arrows: only the arrows from £ are reversed, the arrows from J[ are not reversed. 2.3. For an object A of £ , we denote by C/A the subcategory of £- 2 2 consisting of all morphisms s£C7 with s D = id A , and by H : C/A —> C7 1 A the inclusion functor. Objects of C/A are all morphisms a £ £ with a D = A We write u : a •—^ b if (a,u,Afb) is a morphism of C/A f i«e» a = u b and a D s b D e A in C. • The product of u : a —> b and v : b —> c in C/A is u v : a —^ c . We denote by j/A the full suocategory C/A O C7/JL] of C/A , and by I : j/A —>C/A and H : J/A —^ C-fjl the inclusion functors. Thus I. H A "^ A,JA T , — k-J A A » H IT . Objects of J/A are all morphisms j £ J[ with j D = A • 2,4, fS=f , for f in |C~| = C^ f defines a natural transformation § : D —> D with the following universal property. If /u: P —> F. is a o n 1 ' o 1 natural transfomation of functors F. : K —> C. , then there is exactly one functor M : K -> (£ such that F. » M D. (i « 0, lj and /^ » M <^ • """"•11 * If the functors F. have limits L. * lim F, , with projections L. -—> k F. for k <£|]C| , then there is a unique morphism m ~ li L —>L in £ such that m (kA^) = (k/\ )(kyu) in C. for every 2 One sees easily that m is a limit of the functor M : IC —>(£-• , with project- tions (kAQ> ^A±) : m —>k/u for 2.5. If K is a category and A £| <l| , then a> EA = id A , for all f defines a constant functor E : JC —>£ . If F : ]C —-p C. is a func- tor and fuc : F —> E. a natural transformation, then F=MHAD . /M.= MHAcT ' A A o ' A for exactly one functor M : K -—>G/A . If F has a limit L « lim F , with projections kA: L —>k F for k £ JK\ , then JJL has a limit m « lim yK : L —>A , with ra.« (kA)(kyn) in £ for all k£ JK } . m is also a limit of M in C/A , with projections k A : m —> k^ for k £ JJC 2,6« Let jj be the class of all products u ,j in £ with j £ J, and u isomorphic. We say that jj is left transportable if J, =* J, * The class J^ always is left transportable• Dually, we call J, right transportable in £ if jj is left transportable in £ . 2.7. We recall that a functor T : A —> B has a left adjoint functor S : B. —>4 if and only if there is a natural transformation {*> : Id B. —> S T such that every morphism B ^ 9 B £|B.l , is universal for T , i.e. if A €? |A,| and g : B •—>A T in B. , then g ** (B^g)(f T) in B for exactly one morphism f : B S —>A of 4 . We call yS a front adjunction for T . If A is a subcategory of ]8 and T the inclusion functor, then a univer- sal morphism for T is called a reflection for j^ in B. • Let f & and Pf e ^^^L^l^ - ^ ^ £X C. We say that (pf j) is a strong J-image of f in £ if (a) f « p j in £ and j ^ J. , and (b) when- f f 1 ever f v » u j in £ with j 6J, then u»pxf j v «x j in C for exactly one morphism x^£ . We call (p,j) a J-image of f in £ if (a) is satisfied, and (b) is satisfied for the special case v = f T> . We say that £ has J-dmages» or that £ has strong J-imagea, if every morphism of £ has a jj-image or a strong J^-image respectively in £ . A morphism f££ is called J-extremal or strongly J-extremal if f has a J,-image or strong J.-image (p,j) with j isomorphic in £ • If A€L \Cm\ , then we may define a J-subob.iect of A to be an object of j/A f or of a suit- able skeleton of j/A • Dually, let £ be a class of morphisms of £ . We say that (p,j) is a .> or a strong P-coimage. of f in £ if (j,p) is a Primage, or a op strong Primage, of f in £ . P-coextremal and strongly P-co extremal mor- phisms, and P-quotient objects, are defined accordingly. We usually omit the prefix J[ in proofs and informal discussions. Proposition 3.2. A morphism j of C is in J, if and only if (j T>o9 j) is a strong J-image of j in C fi f f 3.3. It (PJ) i§-£ J-image of f in £ , then (p fj ) is. 1 f 1 a J-image of f in C if and only if j € J.